The grant will help Futamura finish a textbook she is working on about projective geometry and its connection to perspective drawing. Projective geometry is a more abstract form of geometry than the Euclidean geometry students normally learn in high school. It is used to analyze what properties of an object are preserved when you draw or take pictures of it from various angles and has applications for creating movie special effects, video games and augmented reality.

“As an undergraduate I took a projective geometry class the same semester I took a drawing class,” Futamura said. “Ever since then I have wanted to understand the connections between the two.”

While many textbooks have been written on projective geometry for math majors, Futamura said virtually all of them neglect its connections to art.

Futamura is working on the book with Annalisa Crannell, a professor of mathematics at Franklin and Marshall College, and Marc Frantz, a research associate in the Department of Mathematics at Indiana University. Crannell and Frantz previously wrote a textbook on perspective geometry that is widely used in lower-level mathematics courses at liberal arts colleges. Perspective geometry explores the basic mathematics behind drawing in perspective, and requires only knowledge of high school algebra, trigonometry and geometry. Projective geometry dives deeper into the structure of the non-Euclidean geometry implied by projecting the three-dimensional world onto a two-dimensional plane.

The new textbook, which is intended for upper-level students with a strong mathematics background, will have the rigor of a traditional projective geometry textbook but also will include examples from perspective drawing.

Futamura met Crannell and Frantz at a conference where they were giving a mini-course based on their textbook and she asked them about connections between projective geometry and art.

“They had no idea there was this connection,” she said. “This new book compiles all the material we have talked about via email since then.”

So far, the team has eight chapters written for the book. The NSF grant will enable Futamura to continue working on it for the next two summers, and will enable her to bring in a consultant who can help evaluate the effectiveness of the textbook.

“Publishing a textbook requires a lot of testing,” she said.

Futamura is currently testing exercises in the textbook with students in a projective geometry class she is teaching at Southwestern.

“The students are helping point out things I hadn’t considered before,” she said.

Next semester, she will continue testing the exercises with students in a math and art class she is teaching. The grant will enable her to create a website where other teachers can download the exercises to test.

Futamura said the new textbook will help students in a variety of majors. For education majors, it will offer ways to help teach geometry in a more interesting way. It will give art majors a stronger math background and will help computer science majors understand the math behind computer drawings. And it will help math majors draw better and connect abstractness to reality.

The team is already working with Princeton University Press to publish the book. While it could still be five years before the book comes out, Futamura said that does not bother her.